# Affine structures on Lie groupoids

## TL;DR

This paper explores affine structures on Lie groupoids, revealing their rich algebraic and categorical properties, including graded Lie 2-algebra structures and monoidal categories, as a categorification of multiplicative structures.

## Contribution

It introduces the concept of affine structures on Lie groupoids and demonstrates their higher algebraic and categorical frameworks, advancing the understanding of their geometric and algebraic properties.

## Key findings

- Affine structures form a 2-vector space over multiplicative structures.
- The space of affine multivector fields has a graded strict Lie 2-algebra structure.
- Affine (1,1)-tensors form a strict monoidal category.

## Abstract

Affine structures on a Lie groupoid, including affine $k$-vector fields, $k$-forms and $(p,q)$-tensors are studied. We show that the space of affine structures is a 2-vector space over the space of multiplicative structures. Moreover, the space of affine multivector fields has a natural graded strict Lie 2-algebra structure and affine (1,1)-tensors constitute a strict monoidal category. Such higher structures can be seen as the categorification of multiplicative structures on a Lie groupoid.